Let $Y$ be a Fano threefold of index $2$ with $Pic(Y)=\mathbb{Z}$. I understand that $H^0(Y,\mathbb{Z}),H^2(Y,\mathbb{Z}),H^4(Y,\mathbb{Z}),H^6(Y,\mathbb{Z})$ are all $\mathbb{Z}$, generated by the classes of $Y$, a hyperplane, line and point respectively. I am looking for a reference that the Chow ring is also $\mathbb{Z}$ in each degree and generated by the cycle as described above.
I know there are group homomorphisms from $A^k(Y)\to H^{2k}(Y,\mathbb{Z})$ for $k=0,1,2,3$, but struggle to prove why they should be isomorphisms.
Thanks for the help!
Let $\mathrm{CH}^2(X)_{\mathrm{alg}} \subset \mathrm{CH}^2(X)$ be the subgroup of algebraically trivial cycles. Then there is the Abel-Jacobi map $$ \mathrm{CH}^2(X)_{\mathrm{alg}} \to J(X) $$ to the intermediate Jacobian, and the Jacobian is generated by the image. So, if the intermediate Jacobian of $X$ in nontrivial (in the case of Fano varieties of index 2 this happens for $d(X) = H_X^3 \le 4$), the kernel of the cycle class map $$ \mathrm{CH}^2(X) \to H^4(X,\mathbb{Z}) $$ contains the nontrivial subgroup $\mathrm{CH}^2(X)_{\mathrm{alg}}$.